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Theorem dftpos2 7887
Description: Alternate definition of tpos when 𝐹 has relational domain. (Contributed by Mario Carneiro, 10-Sep-2015.)
Assertion
Ref Expression
dftpos2 (Rel dom 𝐹 → tpos 𝐹 = (𝐹 ∘ (𝑥dom 𝐹 {𝑥})))
Distinct variable group:   𝑥,𝐹

Proof of Theorem dftpos2
StepHypRef Expression
1 dmtpos 7882 . . 3 (Rel dom 𝐹 → dom tpos 𝐹 = dom 𝐹)
21reseq2d 5829 . 2 (Rel dom 𝐹 → (tpos 𝐹 ↾ dom tpos 𝐹) = (tpos 𝐹dom 𝐹))
3 reltpos 7875 . . 3 Rel tpos 𝐹
4 resdm 5873 . . 3 (Rel tpos 𝐹 → (tpos 𝐹 ↾ dom tpos 𝐹) = tpos 𝐹)
53, 4ax-mp 5 . 2 (tpos 𝐹 ↾ dom tpos 𝐹) = tpos 𝐹
6 df-tpos 7870 . . . 4 tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}))
76reseq1i 5825 . . 3 (tpos 𝐹dom 𝐹) = ((𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})) ↾ dom 𝐹)
8 resco 6079 . . 3 ((𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})) ↾ dom 𝐹) = (𝐹 ∘ ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ↾ dom 𝐹))
9 ssun1 4127 . . . . 5 dom 𝐹 ⊆ (dom 𝐹 ∪ {∅})
10 resmpt 5881 . . . . 5 (dom 𝐹 ⊆ (dom 𝐹 ∪ {∅}) → ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ↾ dom 𝐹) = (𝑥dom 𝐹 {𝑥}))
119, 10ax-mp 5 . . . 4 ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ↾ dom 𝐹) = (𝑥dom 𝐹 {𝑥})
1211coeq2i 5707 . . 3 (𝐹 ∘ ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ↾ dom 𝐹)) = (𝐹 ∘ (𝑥dom 𝐹 {𝑥}))
137, 8, 123eqtri 2847 . 2 (tpos 𝐹dom 𝐹) = (𝐹 ∘ (𝑥dom 𝐹 {𝑥}))
142, 5, 133eqtr3g 2878 1 (Rel dom 𝐹 → tpos 𝐹 = (𝐹 ∘ (𝑥dom 𝐹 {𝑥})))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  cun 3911  wss 3913  c0 4269  {csn 4543   cuni 4814  cmpt 5122  ccnv 5530  dom cdm 5531  cres 5533  ccom 5535  Rel wrel 5536  tpos ctpos 7869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3753  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-iota 6290  df-fun 6333  df-fn 6334  df-fv 6339  df-tpos 7870
This theorem is referenced by:  tposf12  7895
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